In Exercises 23–30, use expansion by minors to evaluate each determinant. 3 0 02 1 - 52 5 - 1

Verified step by step guidance

1. Identify the matrix as a 3x3 matrix: \( \begin{bmatrix} 3 & 0 & 0 \\ 2 & 1 & -5 \\ 2 & 5 & -1 \end{bmatrix} \).
2. Choose the first row for expansion by minors: \( 3, 0, 0 \).
3. Calculate the determinant of the 2x2 minor matrix for the first element (3): \( \begin{vmatrix} 1 & -5 \\ 5 & -1 \end{vmatrix} \).
4. Calculate the determinant of the 2x2 minor matrix for the second element (0): \( \begin{vmatrix} 2 & -5 \\ 2 & -1 \end{vmatrix} \).
5. Calculate the determinant of the 2x2 minor matrix for the third element (0): \( \begin{vmatrix} 2 & 1 \\ 2 & 5 \end{vmatrix} \).

Key Concepts

Determinants

A determinant is a scalar value that can be computed from the elements of a square matrix. It provides important information about the matrix, such as whether it is invertible and the volume scaling factor of the linear transformation represented by the matrix. The determinant can be calculated using various methods, including expansion by minors.

Expansion by Minors

Expansion by minors is a technique used to calculate the determinant of a matrix by breaking it down into smaller matrices. This method involves selecting a row or column, multiplying each element by its corresponding minor (the determinant of the submatrix formed by removing the row and column of that element), and applying a sign based on the position of the element. This recursive approach simplifies the calculation of determinants for larger matrices.

Cofactors

Cofactors are the signed minors of a matrix element, which are used in the calculation of determinants and in finding the inverse of a matrix. The cofactor of an element is obtained by multiplying its minor by (-1) raised to the power of the sum of the row and column indices of that element. Understanding cofactors is essential for applying the expansion by minors method effectively.